Theorem flimclslem 23897

Description: Lemma for flimcls 23898. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimcls.2	⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))

Assertion

Ref	Expression
flimclslem	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

Proof of Theorem flimclslem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimcls.2	. . 3 ⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
2		topontop 22826	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
4		eqid 2731	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	neisspw 23020	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
6	3, 5	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
7		toponuni 22827	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 = ∪ 𝐽)
9	8	pweqd 4567	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
10	6, 9	sseqtrrd 3972	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
11		toponmax 22839	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		elpw2g 5271	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
14	13	biimpar 477	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
15	14	3adant3 1132	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
16	15	snssd 4761	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ⊆ 𝒫 𝑋)
17	10, 16	unssd 4142	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋)
18		ssun2 4129	. . . . . 6 ⊢ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})
19	11	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 ∈ 𝐽)
20		simp2 1137	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
21	19, 20	ssexd 5262	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
22	21	snn0d 4728	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ≠ ∅)
23		ssn0 4354	. . . . . 6 ⊢ (({𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∧ {𝑆} ≠ ∅) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
24	18, 22, 23	sylancr 587	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
25	20, 8	sseqtrd 3971	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
26		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
27	4	neindisj 23030	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ (𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑆) ≠ ∅)
28	27	expr 456	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
29	3, 25, 26, 28	syl21anc 837	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
30	29	imp 406	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥 ∩ 𝑆) ≠ ∅)
31		elsni 4593	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑆} → 𝑦 = 𝑆)
32	31	ineq2d 4170	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑆))
33	32	neeq1d 2987	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑆} → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑥 ∩ 𝑆) ≠ ∅))
34	30, 33	syl5ibrcom 247	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) ≠ ∅))
35	34	ralrimiv 3123	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
36	35	ralrimiva 3124	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
37		simp1 1136	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘𝑋))
38	4	clsss3 22972	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
39	3, 25, 38	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
40	39, 26	sseldd 3935	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ∪ 𝐽)
41	40, 8	eleqtrrd 2834	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ 𝑋)
42	41	snssd 4761	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ⊆ 𝑋)
43		snnzg 4727	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → {𝐴} ≠ ∅)
44	43	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ≠ ∅)
45		neifil 23793	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	37, 42, 44, 45	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
47		filfbas 23761	. . . . . . . 8 ⊢ (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
48	46, 47	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
49		ne0i 4291	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → ((cls‘𝐽)‘𝑆) ≠ ∅)
50	49	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
51		cls0 22993	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
52	3, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘∅) = ∅)
53	50, 52	neeqtrrd 3002	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅))
54		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑆 = ∅ → ((cls‘𝐽)‘𝑆) = ((cls‘𝐽)‘∅))
55	54	necon3i 2960	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅) → 𝑆 ≠ ∅)
56	53, 55	syl 17	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ≠ ∅)
57		snfbas 23779	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ∧ 𝑋 ∈ 𝐽) → {𝑆} ∈ (fBas‘𝑋))
58	20, 56, 19, 57	syl3anc 1373	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ∈ (fBas‘𝑋))
59		fbunfip 23782	. . . . . . 7 ⊢ ((((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋) ∧ {𝑆} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
60	48, 58, 59	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
61	36, 60	mpbird 257	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
62		fsubbas 23780	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
63	19, 62	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
64	17, 24, 61, 63	mpbir3and 1343	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋))
65		fgcl 23791	. . . 4 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
66	64, 65	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
67	1, 66	eqeltrid 2835	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐹 ∈ (Fil‘𝑋))
68		fvex 6835	. . . . . 6 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
69		snex 5374	. . . . . 6 ⊢ {𝑆} ∈ V
70	68, 69	unex 7677	. . . . 5 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V
71		ssfii 9303	. . . . 5 ⊢ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
72	70, 71	ax-mp 5	. . . 4 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
73		ssfg 23785	. . . . . 6 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
74	64, 73	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
75	74, 1	sseqtrrdi 3976	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ 𝐹)
76	72, 75	sstrid 3946	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝐹)
77		snssg 4736	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
78	21, 77	syl 17	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
79	18, 78	mpbiri 258	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
80	76, 79	sseldd 3935	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐹)
81	76	unssad 4143	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
82		elflim 23884	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
83	37, 67, 82	syl2anc 584	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
84	41, 81, 83	mpbir2and 713	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ (𝐽 fLim 𝐹))
85	67, 80, 84	3jca 1128	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  Vcvv 3436   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859  ‘cfv 6481  (class class class)co 7346  ficfi 9294  fBascfbas 21277  filGencfg 21278  Topctop 22806  TopOnctopon 22823  clsccl 22931  neicnei 23010  Filcfil 23758   fLim cflim 23847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1o 8385  df-2o 8386  df-en 8870  df-fin 8873  df-fi 9295  df-fbas 21286  df-fg 21287  df-top 22807  df-topon 22824  df-cld 22932  df-ntr 22933  df-cls 22934  df-nei 23011  df-fil 23759  df-flim 23852

This theorem is referenced by:  flimcls  23898